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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 121275bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121275.bu2 | 121275bg1 | \([1, -1, 1, -18605, -176228]\) | \(19683/11\) | \(398008405265625\) | \([2]\) | \(345600\) | \(1.4917\) | \(\Gamma_0(N)\)-optimal |
121275.bu1 | 121275bg2 | \([1, -1, 1, -183980, 30252772]\) | \(19034163/121\) | \(4378092457921875\) | \([2]\) | \(691200\) | \(1.8383\) |
Rank
sage: E.rank()
The elliptic curves in class 121275bg have rank \(1\).
Complex multiplication
The elliptic curves in class 121275bg do not have complex multiplication.Modular form 121275.2.a.bg
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.